A Generalized Mode-Matching Technique in the Theory of Waveguide Mode Diffraction. Part 4: Convergence Rate of the Projection Approximations”

Abstract

This part of the study continues presenting fundamentals of the generalized mode-matching technique designed for analyzing the waveguide mode scattering. Specifically, the problem is considered of analytically estimating the rate of convergence of the projection approximations to the operator Fresnel formulas the unconditional convergence of which has been proven earlier. By the way of example of a canonical scalar problem of wave diffraction by a step in a rectangular waveguide the derivation of the approximation error of the wave reflection and transmission operators is presented. It is shown that the formulated problem can be solved through considering the strong P-convergence of the projection representation of the amplitude scattering operator. As a result, an analytical estimate has been first obtained for the rate of convergence of the scattering operator approximations obtained using the truncation method for the operator Fresnel formulas. The found regularities are validated through numerical calculations. The obtained results make it possible to determine the computational efficiency of the generalized mode-matching technique.